Exponential Functions

Constant ratio: the defining property

Here is the distinction that matters.

A linear function $f(x) = mx + b$ has a constant difference between outputs: $f(x+1) - f(x) = m$, regardless of where you are on the $x$-axis.

An exponential function $f(x) = a \cdot b^x$ has a constant ratio between outputs: $\frac{f(x+1)}{f(x)} = \frac{a \cdot b^{x+1}}{a \cdot b^x} = b.$

Every time $x$ increases by $1$, you multiply the output by $b$. That's it. $b$ is called the base, and it's the whole engine.

This matters because multiplication compounds and it does not stop. After $n$ steps, starting from $a$, you have $a \cdot b^n$. After $n$ steps of linear growth from $a$ with rate $m$, you have $a + mn$. The linear function adds $mn$ total; the exponential multiplies $b^n$ times. For $b > 1$ and large $n$, these are in completely different universes — one of them is still in the parking lot while the other is orbiting Jupiter.

The graph of $b^x$

Let $f(x) = b^x$ with $b > 0$, $b \ne 1$.

Growth: $b > 1$.

• Always positive: $b^x > 0$ for all $x$.
• Passes through $(0, 1)$ always: $b^0 = 1$.
• As $x \to +\infty$: $f(x) \to +\infty$ (steep growth).
• As $x \to -\infty$: $f(x) \to 0^+$ (approaches the $x$-axis from above, never touching it).
• Horizontal asymptote: $y = 0$.

Decay: $0 < b < 1$.

• Same shape, reflected horizontally. Now: as $x \to +\infty$, $f(x) \to 0^+$; as $x \to -\infty$, $f(x) \to +\infty$.
• Same horizontal asymptote $y = 0$.

Drag the grapher to see growth ($b = 2$) and decay ($b = 1/2$) side by side:

Compound interest: the canonical story

Suppose you deposit principal $P$ in an account paying annual interest rate $r$ (as a decimal), compounded $n$ times per year. After $t$ years, the balance is: $A = P\!\left(1 + \frac{r}{n}\right)^{nt}.$

Why? After one compounding period, you have $P \cdot (1 + r/n)$. After two periods, $P \cdot (1+r/n)^2$. After $nt$ total periods, $P \cdot (1+r/n)^{nt}$. This is the constant-ratio story in action: each period multiplies by $(1 + r/n)$.

Half-life: the canonical decay story. A radioactive substance with half-life $H$ has amount: $A(t) = A_0 \cdot \left(\frac{1}{2}\right)^{t/H} = A_0 \cdot 2^{-t/H}.$

After every $H$ time units, the amount halves — constant ratio $1/2$ per $H$ years.

Exponentials vs polynomials: the massacre

Here is a numerical demonstration. I want you to watch carefully because the Federation of Boring Textbook Authors either skips this or buries it in a footnote, and that is a crime. Let $f(x) = x^{100}$ (a ferocious polynomial) and $g(x) = 2^x$ (a humble exponential with the smallest non-trivial integer base).

The polynomial is winning massively at $x = 10$. By $x = 1000$ it's a tie. By $x = 10000$, the exponential has lapped the polynomial so many times it's embarrassing. The precise statement: for any polynomial $p(x)$ and any base $b > 1$,

$\lim_{x \to \infty} \frac{p(x)}{b^x} = 0.$

This is a calculus result (L'Hopital repeated application), but the numerical table makes it undeniable. Exponentials eventually crush every polynomial. No exceptions, no negotiations. File this away; it will matter in computer science (time complexity), finance, biology, and physics, and anyone who tells you otherwise is selling you bullshit.

The number $e$: the natural base

There is one base that is mathematically natural in a way no other base is. Consider the question: for what base $b$ does the exponential function $b^x$ have the property that its *rate of change at $x = 0$ equals exactly $1$?

Answer: $b = e$, where: $e = \lim_{n \to \infty}\!\left(1 + \frac{1}{n}\right)^n \approx 2.71828\ldots$

This is irrational (and in fact transcendental — a fact that requires significant machinery to prove). You can think of $e$ as the continuous-compounding limit: if you compound interest continuously instead of finitely many times per year, the $1$ dollar balance grows to $e^r$ dollars after one year at rate $r$.

The exact reason $e$ is natural — that $\frac{d}{dx}e^x = e^x$ (the derivative of $e^x$ is itself) — requires calculus. But the number $e$ and the function $e^x = \exp(x)$ appear now because you need them for logarithms in the next lesson. You will see $e$ in every stratum from here to the spectral theorem, lurking in every corner of mathematics like a brilliant and slightly unhinged colleague who shows up uninvited to every party and is always right. Treat it with respect.